Minimax算法与Alpha-Beta算法

博弈树
~描述：节点是状态，边表示动作或落子。
~原则：己方利益最大化，对方利益最小化。
Minimax算法
~定义:在理想对手的假设下，MIN方总是最小化MAX方的最大化努力。
~原理：使用了简单的深度递归搜索技术来确定每一个节点的值：它从根节点开始，一直前进到叶子节点，然后在递归回溯时，将叶子节点的效用值往上回传——对于MAX方，计算最大值，对于MIN方，计算最小值。
~示例：以下是一个2层（Two-Ply）博弈树：
1.叶子节点的效用值越大，对MAX方越有利。在MIN方会选择该节点下效用值最小的节点。即 3，2，2（第三层从左向右依次）。
2.MAX方要在“3，2，2”选择一个，选择对MAX方最有利的3。

~注：上图有颜色的方框是用Alpha-Beta算法分析的，用于将两种算法进行对比，下面将会对该算法扩展分析。

Alpha-Beta算法
以下是将上面的博弈树用Alpha-Beta算法的分解分析：
(a).

(b).

©.

(d).

(e).

(f).

总结分析：
Alpha-Beta算法用了剪枝而Minimax算法没有，从代码的运行可以明显看出二者速度的差别。

Minimax算法的代码：

# -*- coding: utf-8 -*-
"""
Created on Fri Oct 26 14:41:12 2018@author: duxiaoqin
Functions:(1) Minimax Algorithm for TicTacToe
"""from graphics import *
from tictactoe import *
from tttdraw import *
from tttinput import *
import sysdef Minimax(node, depth):result = node.isGameOver()if result != None:return result, (), depthif node.getPlayer() == TicTacToe.BLACK:bestValue = -sys.maxsizebestMove = ()bestDepth = sys.maxsizemoves = node.getAllMoves()for move in moves:child = node.clone()child.play(*move)v, _, leafDepth = Minimax(child, depth+1)if bestValue == v and bestDepth > leafDepth:bestValue = vbestMove = movebestDepth = leafDepthif bestValue < v:bestValue = vbestMove = movebestDepth = leafDepthreturn bestValue, bestMove, bestDepthelse:bestValue = sys.maxsizebestMove = ()bestDepth = sys.maxsizemoves = node.getAllMoves()for move in moves:child = node.clone()child.play(*move)v, _, leafDepth = Minimax(child, depth+1)if bestValue == v and bestDepth > leafDepth:bestValue = vbestMove = movebestDepth = leafDepthif bestValue > v:bestValue = vbestMove = movebestDepth = leafDepthreturn bestValue, bestMove, bestDepthdef main():win = GraphWin('Minimax for TicTacToe', 600, 600, autoflush=False)ttt = TicTacToe()tttdraw = TTTDraw(win)tttinput = TTTInput(win)tttdraw.draw(ttt)while win.checkKey() != 'Escape':if ttt.getPlayer() == TicTacToe.WHITE:v, move, _ = Minimax(ttt, 0)if move != ():ttt.play(*move)tttinput.input(ttt)tttdraw.draw(ttt)if ttt.isGameOver() != None:time.sleep(1)ttt.reset()tttdraw.draw(ttt)#win.getMouse()win.close()if __name__ == '__main__':main()

Alpha-Beta算法的代码：

# -*- coding: utf-8 -*-
"""
Created on Fri Oct 26 20:53:08 2018@author: duxiaoqin
Functions:(1) Alpha-Beta Algorithm for TicTacToe
"""from graphics import *
from tictactoe import *
from tttdraw import *
from tttinput import *
import sys
import timedef AlphaBeta(node, depth, alpha, beta):result = node.isGameOver()if result != None:#游戏结束return result, (), depthif node.getPlayer() == TicTacToe.BLACK:bestValue = -sys.maxsizebestMove = ()bestDepth = sys.maxsizemoves = node.getAllMoves()for move in moves:child = node.clone()child.play(*move)v, _, leafDepth = AlphaBeta(child, depth+1, alpha, beta)if bestValue == v and bestDepth > leafDepth:bestValue = vbestMove = movebestDepth = leafDepthif bestValue < v:bestValue = vbestMove = movebestDepth = leafDepthalpha = max(alpha, bestValue)if beta <= alpha:#进行剪枝break #beta pruningreturn bestValue, bestMove, bestDepthelse:bestValue = sys.maxsizebestMove = ()bestDepth = sys.maxsizemoves = node.getAllMoves()for move in moves:child = node.clone()child.play(*move)v, _, leafDepth = AlphaBeta(child, depth+1, alpha, beta)#‘_’在该函数内不起作用if bestValue == v and bestDepth > leafDepth:bestValue = vbestMove = movebestDepth = leafDepthif bestValue > v:bestValue = vbestMove = movebestDepth = leafDepthbeta = min(beta, bestValue)if beta <= alpha:break #alpha pruningreturn bestValue, bestMove, bestDepthdef main():win = GraphWin('Minimax for TicTacToe', 600, 600, autoflush=False)ttt = TicTacToe()tttdraw = TTTDraw(win)tttinput = TTTInput(win)tttdraw.draw(ttt)while win.checkKey() != 'Escape':if ttt.getPlayer() == TicTacToe.WHITE:v, move, _ = AlphaBeta(ttt, 0, -sys.maxsize, sys.maxsize)if move != ():ttt.play(*move)tttinput.input(ttt)tttdraw.draw(ttt)if ttt.isGameOver() != None:time.sleep(1)ttt.reset()tttdraw.draw(ttt)#win.getMouse()win.close()if __name__ == '__main__':main()

博弈树代码：

# -*- coding: utf-8 -*-
"""
Created on Mon Sep 10 22:25:08 2018@author: duxiaoqinFunctions:(1) TicTacToe class;
"""from myarray2d import Array2Dclass TicTacToe:BLACK = TrueWHITE = FalseEMPTY = NoneBLACKWIN = 1WHITEWIN = -1DRAW = 0def __init__(self):self.board = Array2D(3, 3)self.player = TicTacToe.BLACKself.black = []self.white = []self.magic = Array2D(3, 3)self.magic[0, 0] = 2self.magic[0, 1] = 9self.magic[0, 2] = 4self.magic[1, 0] = 7self.magic[1, 1] = 5self.magic[1, 2] = 3self.magic[2, 0] = 6self.magic[2, 1] = 1self.magic[2, 2] = 8def reset(self):self.board.clear(None)self.player = TicTacToe.BLACKself.black = []self.white = []def clone(self):newttt = TicTacToe()for row in range(3):for col in range(3):newttt.board[row, col] = self.board[row, col]newttt.player = self.playernewttt.black = self.black[:]newttt.white = self.white[:]return newtttdef ToString(self):l = []for row in range(3):for col in range(3):if self.board[row, col] == TicTacToe.BLACK:l.append('X')elif self.board[row, col] == TicTacToe.WHITE:l.append('O')else:l.append('_')return ''.join(l)def print(self):for row in range(3):for col in range(3):if self.board[row, col] == TicTacToe.BLACK:print('X', end=' ')elif self.board[row, col] == TicTacToe.WHITE:print('O', end=' ')else:print('_', end=' ')print()def play(self, row, col):self.board[row, col] = self.playerif self.player == TicTacToe.BLACK:self.black.append(self.magic[row, col])else:self.white.append(self.magic[row, col])self.player = not self.playerdef getPlayer(self):return self.playerdef getAllMoves(self):return [(row, col) for row in range(3) \for col in range(3) \if self.board[row, col] == TicTacToe.EMPTY]def isWin(self, n, goal, moves):moves_clone = moves[:]if n == 0:return goal == 0elif goal <= 0:return Falseelif len(moves_clone) == 0:return Falseelse:item = moves_clone.pop(0)if self.isWin(n-1, goal-item, moves_clone[:]):return Trueelif self.isWin(n, goal, moves_clone[:]):return Truereturn Falsedef isGameOver(self):if self.isWin(3, 15, self.black):return TicTacToe.BLACKWINelif self.isWin(3, 15, self.white):return TicTacToe.WHITEWINelif len(self.black)+len(self.white) == 9:return TicTacToe.DRAWelse:return Nonedef main():ttt = TicTacToe()ttt.play(1, 1)ttt.play(0, 0)ttt.play(2, 0)ttt.play(0, 1)ttt.play(0, 2)ttt.print()print(ttt.isGameOver())print(ttt.ToString())if __name__ == '__main__':main()

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